Cooperation-based optimization of industrial supply chains

Transcription

1 Cooperation-based optimization of industrial supply chains James B. Rawlings, Brett T. Stewart, Kaushik Subramanian and Christos T. Maravelias Department of Chemical and Biological Engineering May 9 2, 2 Workshop on Distributed Model Predictive Control and Supply Chains Lund Center for Control of Complex Engineering Systems Lund University Rawlings Optimization of supply chains / 32

2 Outline Overview of Distributed Model Predictive Control Control of large-scale systems Stability theory for cooperative MPC 2 Challenges for Cooperative MPC of Supply Chains 3 Conclusions and Future Outlook Rawlings Optimization of supply chains 2 / 32

3 Predictive control Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u min u(t) T y sp g(x, u) 2 Q + u sp u 2 R dt ẋ = f (x, u) x() = x (given) y = g(x, u) Rawlings Optimization of supply chains 3 / 32

4 State estimation Measurement MH Estimate MPC control Forecast t time Reconcile the past Forecast the future sensors y actuators u min x,w(t) T y g(x, u) 2 R + ẋ f (x, u) 2 Q dt ẋ = f (x, u) + w (process noise) y = g(x, u) + v (measurement noise) Rawlings Optimization of supply chains 4 / 32

5 Electrical power distribution Rawlings Optimization of supply chains 5 / 32

6 Chemical plant integration Material flow Energy flow Rawlings Optimization of supply chains 6 / 32

7 MPC at the large scale Decentralized Control Most large-scale systems consist of networks of interconnected/interacting subsystems Chemical plants, electrical power grids, water distribution networks,... Traditional approach: Decentralized control Wealth of literature from the early 97 s on improved decentralized control a Well known that poor performance may result if the interconnections are not negligible a (Sandell Jr. et al., 978; Šiljak, 99; Lunze, 992) Rawlings Optimization of supply chains 7 / 32

8 MPC at the large scale Centralized Control Steady increase in available computing power has provided the opportunity for centralized control Coordinated control: Distributed optimization to achieve fast solution of centralized control (Necoara et al., 28; Cheng et al., 27) Most practitioners view centralized control of large, networked systems as impractical and unrealistic A divide and conquer strategy is essential for control of large, networked systems (Ho, 25) Centralized control: A benchmark for comparing and assessing distributed controllers Rawlings Optimization of supply chains 8 / 32

9 Nomenclature: consider two interacting units Objective functions V (u, u 2 ), V 2 (u, u 2 ) and V (u, u 2 ) = w V (u, u 2 ) + w 2 V 2 (u, u 2 ) decision variables for units u Ω, u 2 Ω 2 Decentralized Control Noncooperative Control (Nash equilibrium) Cooperative Control (Pareto optimal) Centralized Control (Pareto optimal) min Ṽ (u ) u Ω min V (u, u 2 ) u Ω min V (u, u 2 ) u Ω min Ṽ 2 (u 2 ) u 2 Ω 2 min V 2 (u, u 2 ) u 2 Ω 2 min V (u, u 2 ) u 2 Ω 2 min V (u, u 2 ) u,u 2 Ω Ω 2 Rawlings Optimization of supply chains 9 / 32

10 Noninteracting systems 2 b V 2 (u) n, d, p u 2 a - V (u) u Rawlings Optimization of supply chains / 32

11 Weakly interacting systems.5 V 2 (u) b p n, d -.5 u 2 - a V (u) u Rawlings Optimization of supply chains / 32

12 Moderately interacting systems 2.5 V (u) u 2.5 b V 2 (u) p a d n u Rawlings Optimization of supply chains 2 / 32

13 Strongly interacting (conflicting) systems 2.5 V (u).5 V 2 (u) p a u 2 b d u Rawlings Optimization of supply chains 3 / 32

14 Strongly interacting (conflicting) systems 6 4 n 2 u V 2 (u) V (u) u Rawlings Optimization of supply chains 4 / 32

15 Geometry of cooperative vs. noncooperative MPC p n 5 a u 2 V (u) V 2 (u) b u Rawlings Optimization of supply chains 5 / 32

16 Plantwide suboptimal MPC n p 5 a u 2 V (u) V 2 (u) b u Early termination of optimization gives suboptimal plantwide feedback Use suboptimal MPC theory to prove stability Rawlings Optimization of supply chains 6 / 32

17 Plantwide suboptimal MPC Consider closed-loop system augmented with input trajectory ( ) ( ) x + Ax + Bu u + = g(x, u) Function g( ) returns suboptimal choice Stability of augmented system is established by Lyapunov function a (x, u) 2 V (x, u) b (x, u) 2 V (x +, u + ) V (x, u) c (x, u) 2 Adding constraint establishes closed-loop stability of the origin for all u u d x x B r, r > Cooperative optimization satisfies these properties for plantwide objective function V (x, u) (Rawlings and Mayne, 29, pp.48-42) Rawlings Optimization of supply chains 7 / 32

18 Modeling Plantwide step response u u 2 y y 2 Interaction models found by decentralized identification 2 u y x + = A x + B u y 2 x + 2 = A 2x 2 + B 2 u 2 Gudi and Rawlings (26) Rawlings Optimization of supply chains 8 / 32

19 Modeling Consider the linearized physical model x + = Ax + B u + B 2 u 2 y = C x, y 2 = C 2 x Kalman canonical form of the triple (A, B j, C i ) zij oc zōc ij zij o c zō c ij + = Interaction models A oc ij A oc c Aōoc ij Aōc ij A o c Aō co y ij = [ C oc ij C o c A ij A oc ij ij Aōco c ij Aōc c ij ij ij Aō c ij zij oc ij ] B ij B oc ij zōc ij zij o c zō c ij C ij C oc ij zij oc zōc ij zij o c zō c ij + y i = j Bij oc Bōc ij y ij x ij z oc ij u j Rawlings Optimization of supply chains 9 / 32

20 Unstable modes For unstable systems, we zero the unstable modes with terminal constraints. For subsystem S u x (N) = S u 2 x 2 (N) = To ensure terminal constraint feasibility for all x, we require (A, B ) stabilizable [ ] [ ] A B A = B = A 2 B 2 For output feedback, we require (A, C ) detectable [ ] A A = C = [ ] C C 2 A 2 Similar requirements for other subsystem Rawlings Optimization of supply chains 2 / 32

21 Output feedback Consider augmented system perturbed by stable estimator ˆx + Aˆx + Bu + Le u = e + g(ˆx, u, e) A L e Stable estimator error implies Lyapunov function ā e J(e) b e J(e + ) J(e) c e Stability of perturbed system established by Lyapunov function W (ˆx, u, e) = V (ˆx, u) + J(e) Rawlings Optimization of supply chains 2 / 32

22 Two reactors with separation and recycle D, x Ad, x Bd MPC 3 MPC MPC 2 F purge F, x A F, x A H r Hm F m, x Am, x Bm H b Q F r, x Ar, x Br A B B C A B B C F b, x Ab, x Bb, T Rawlings Optimization of supply chains 22 / 32

23 Two reactors with separation and recycle H m Time H b Time setpoint Cent-MPC Ncoop-MPC Coop-MPC ( iterate) setpoint Cent-MPC Ncoop-MPC Coop-MPC ( iterate) F Time Cent-MPC Ncoop-MPC Coop-MPC ( iterate) D Time Cent-MPC Ncoop-MPC Coop-MPC ( iterate) Rawlings Optimization of supply chains 23 / 32

24 Two reactors with separation and recycle Performance comparison Cost ( 2 ) Performance loss Centralized MPC.75 Decentralized MPC Noncooperative MPC Cooperative MPC ( iterate) % Cooperative MPC ( iterates).84 5% Rawlings Optimization of supply chains 24 / 32

25 Cooperative MPC of supply chains Previous work on supply chain modeling and optimization 3 Inventories and backorders are subsystem states Downstream product shipments and upstream orders are subsystem inputs Inventories and backorders modeled as integrators (tanks) Stabilizability and detectability assumptions not satisfied [ ] [ ] I Bi A i = B I i = B 2i [ ] I A i = C I i = [ ] C i C i2 Implementation of cooperative MPC for supply chains remains a challenge 3 Perea López et al. (23); Mestan et al. (26); Braun et al. (22); Seferlis and Giannelos (24) Rawlings Optimization of supply chains 25 / 32

26 Cooperative MPC of supply chains Possible solution I: Coupled constraints Work with minimal (A, B, C) supply chain model Terminal constraint S u x(n) = coupled in subsystem inputs Challenge Cooperative optimization does not converge to Pareto optimum with coupled constraints Share coupled inputs among subsystems to achieve Pareto optimal performance In the limit of full supply chain coupling, each subsystem solves the centralized optimization Alternative To avoid centralized optimization, share inputs with only nearest neighbors for near optimal performance Rawlings Optimization of supply chains 26 / 32

27 Cooperative MPC of supply chains Possible solution II: Centralized estimation (A i, B i ) not stabilizable, but there is a stabilizable subspace X i X i = { x i u i : [ A n ] i B i B i ui = A n } i x i Any x i X i can be brought to the origin Challenge Must ensure estimated states are in stabilizable subspace Estimation must be centralized Trade-offs No coupled constraints, therefore cooperative optimization converges to Pareto optimum Easy to enforce x i X i Subsystems must share output measurements Supply chain subsystems cannot choose estimators independently Rawlings Optimization of supply chains 27 / 32

28 Conclusions Cooperative MPC theory maturing a satisfies hard input constraints provides nominal stability for plants with even strongly interacting subsystems retains closed-loop stability for early iteration termination converges to Pareto optimal control in the limit of iteration remains stable under perturbation from stable state estimator avoids coordination layer Cooperative MPC for supply chains remains a challenge stabilizability and detectability assumptions not satisfied many alternative solution strategies exist each strategy has drawbacks a?maestre et al. (2) Rawlings Optimization of supply chains 28 / 32

29 Future directions Supply chains Evaluate alternative supply chain cooperative control strategies Industrial application: gas supplier (Praxair), steel mill, power utility Cooperative MPC Hierarchical implementation a time scale separation delayed communication reduced information sharing optimization at MPC layer only Nonlinear models a Stewart et al. (2) Rawlings Optimization of supply chains 29 / 32

30 MPC Monograph Chapter 6 on distributed MPC 576 page text 24 exercises 335 page solution manual 3 appendices on web (33 pages) Rawlings Optimization of supply chains 3 / 32

31 Further reading I M. W. Braun, D. E. Rivera, W. M. Carlyle, and K. G. Kempf. A model predictive control framework for robust management of multi-product, multi-echelon demand networks. In IFAC, 5th Triennial World Congress, 22. R. Cheng, J. F. Forbes, and W. S. Yip. Price-driven coordination method for solving plant-wide MPC problems. J. Proc. Cont., 7(5): , 27. R. D. Gudi and J. B. Rawlings. Identification for Decentralized Model Predictive Control. AIChE J., 52(6):298 22, 26. Y.-C. Ho. On Centralized Optimal Control. IEEE Trans. Auto. Cont., 5(4): , 25. J. Lunze. Feedback Control of Large Scale Systems. Prentice-Hall, London, U.K., 992. J. M. Maestre, D. Muñoz de la Peña, and E. F. Camacho. Distributed model predictive control based on a cooperative game. Optimal Cont. Appl. Meth., In press, 2. E. Mestan, M. Türkay, and Y. Arkun. Optimization of operations in supply chain systems using hybrid systems approach and model predictive control. Ind. Eng. Chem. Res., 45: , August 26. Rawlings Optimization of supply chains 3 / 32

32 Further reading II I. Necoara, D. Doan, and J. Suykens. Application of the proximal center decomposition method to distributed model predictive control. In Proceedings of the IEEE Conference on Decision and Control, Cancun, Mexico, December E. Perea López, B. E. Ydstie, and I. E. Grossmann. A model predictive control strategy for supply chain optimization. Comput. Chem. Eng., 27(8-9):2 28, February 23. J. B. Rawlings and D. Q. Mayne. Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI, pages, ISBN N. R. Sandell Jr., P. Varaiya, M. Athans, and M. Safonov. Survey of decentralized control methods for large scale systems. IEEE Trans. Auto. Cont., 23(2):8 28, 978. P. Seferlis and N. F. Giannelos. A two-layered optimization-based control strategy for multi-echelon supply chain networks. Comput. Chem. Eng., 28:2 29, 24. D. D. Šiljak. Decentralized Control of Complex Systems. Academic Press, London, 99. ISBN B. T. Stewart, J. B. Rawlings, and S. J. Wright. Hierarchical cooperative distributed model predictive control. In Proceedings of the American Control Conference, Baltimore, Maryland, June 2. Rawlings Optimization of supply chains 32 / 32